1Department of Biological Sciences, University of Illinois, Chicago, IL 60607, 2Department of Fisheries and Wildlife, University of Minnesota, St. Paul, MN 55108 and 3Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA

Author to whom all correspondence should be addressed. e-mail: squirrel@uic.edu

ABSTRACT

Question: We examine strategy dynamics for evolutionary games with vector-valued strategies.

Mathematical methods: We use the fitness-generating function (G-function) to derive ESS maximum principle and strategy dynamics along multi-dimensional adaptive landscapes. We apply the dynamics to two models of co-evolution of competitors and one model of predator–prey co-evolution.

Results: When several traits evolve simultaneously (vector-valued strategies), a trait changes according to the product of each trait’s fitness gradient and the matrix of covariances between the trait and the other traits. Only when these covariance terms are small does each trait in the strategy vector change in the direction of that trait’s fitness gradient, similar to the scalar case. In the models, convergent stable points may be minima, maxima or saddle points. And the ESS may ultimately contain one, two or many species at distinct peaks of the adaptive landscape. The predator–prey model illustrates how the ESS strategy for the predator can result in an ESS for the prey that allows for any number of prey species to exist along a (flat) rim of a crater in the prey’s adaptive landscape.

Key conclusions: In going from a scalar to a vector-valued strategy, the adaptive dynamics become more complicated with respect to fitness gradients, the bestiary associated with convergent stable points increases, the avenues for adaptive speciation and achieving an ESS increase, and the number of non-ESS species that can co-exist ecologically increases. By increasing the dimensionality of the adaptive surface, vector-valued strategies increase the opportunity for frequency-dependent selection to produce the number of species and the appropriate trait combinations for an ESS.

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